Optical Anisotropy of Nanotube Suspensions Erik K

JOURNAL OF CHEMICAL PHYSICS VOLUME 121, NUMBER 2 8 JULY 2004

Optical anisotropy of nanotube suspensions Erik K. Hobbiea) National Institute of Standards and Technology, Gaithersburg, Maryland 20899 ͑Received 16 January 2004; accepted 7 April 2004͒ A semimacroscopic model of an optically anisotropic nanotube suspension is derived perturbatively from Maxwell’s equations in a dielectric medium. We calculate leading-order expressions, valid in the dilute and semidilute limits, for the intrinsic and form contributions to the complex dielectric tensor in terms of the volume fraction, mean orientation, aspect ratio, optical anisotropy, and optical contrast of the nanotubes. The birefringence and dichroism are derived explicitly to leading order in ﬂuctuations, and the connection with depolarized light scattering is established. The results are generalized to include tube deformation. © 2004 American Institute of Physics. ͓DOI: 10.1063/1.1760079͔

I. INTRODUCTION sions and melts. The approach is reminiscent of the general theory of flow-induced optical anisotropy in dilute polymer It is well known that flow can induce optical anisotropy solutions derived by Onuki and Doi,6 and it expands on a in macromolecular fluids and colloidal suspensions in the previous treatment of depolarized light scattering from car- 1,2 form of birefringence and dichroism. As first demonstrated bon nanotube suspensions.5 We derive general expressions 3 by Peterlin and Stewart, these effects separate naturally into for both the intrinsic and form contributions to the dielectric two distinct parts; an intrinsic contribution arising from the tensor, and we derive leading-order expressions for the bire- orientation of optically anisotropic polymer segments or par- fringence and dichroism. The connection to depolarized light ticles, and a form contribution arising from anisotropy in the scattering is established and the results are generalized to segmental or density correlation function. The experimental include tube deformation. techniques of flow birefringence and flow dichroism have proven to be powerful tools for studying the structural and dynamic properties of flowing polymer melts, polymer solu- II. INTRINSIC CONTRIBUTION tions, and anisotropic suspensions, which in turn has contrib- For a quasistatic suspension of N optically anisotropic uted to recent technological advances in the engineering and nanotubes in an optically isotropic solvent of dielectric con- ␧ flow processing of complex fluids. stant s illuminated by an incident electric field E(r,t) ϭ i(k rϪ␻t) 2ϭ␧ ␻2 2 Recently, carbon nanotubes have emerged as novel ma- E0e • , where k s /c and k points along the terials with potential applications in a host of advanced direction of propagation ͑see Fig. 1͒, the macroscopic dielec- 4 technologies. Nanocomposites engineered from polymers tric tensor can be defined via and carbon nanotubes, for example, offer the promise of ͑ ͒ϭ␧ ͑ ͒ϭ␧ i(k rϪ␻t) ͑ ͒ plastic materials with enhanced structural, thermal, elec- D r,t ˜ •E r,t ˜ •E0e • , 1 ␧ϭ͚3 ␧ tronic, and optical properties. Efficient bulk processing of where the tensor ˜ ␮,␯ϭ1xˆ␮ ␮␯xˆ␯ is homogeneous. If r such composites will depend in part on a detailed under- locates a segment of a nanotube, nˆ(r), then the spatially standing of the response of carbon nanotube suspensions and varying amplitude of the displacement field induced at r is melts to flow. As colloidal particles, nanotubes are also D͑r͒ϭ͕␣ʈnˆ͑r͒nˆ͑r͒ϩ␣Ќ͓1Ϫnˆ͑r͒nˆ͑r͔͖͒ E , ͑2͒ somewhat unique and intriguing, in that they straddle the • 0

interface between semiflexible polymers and microscopic fi- where the local complex dielectric constants ␣ʈ and ␣Ќ arise bers. Given the significant optical contrast between carbon from polarization and absorption along and normal to the nanotubes and most typical polymers, it is reasonable to ex- local symmetry axis of the nanotube, respectively, and 1 is

pect that such suspensions will exhibit reasonably strong op- the identity tensor. The parameters ␣ʈ and ␣Ќ appear in the tical anisotropy, making them well-suited to rheo-optical formulas for the depolarized scattering of monochromatic techniques.5 Applied to such novel soft materials, the meth- light from nanotube suspensions.5 Equation ͑2͒ defines a ods of flow birefringence and flow dichroism might offer local dielectric tensor as new physical insight into both fundamental flow-structure ˜␧͑r͒ϭ␣ʈnˆ͑r͒nˆ͑r͒ϩ␣Ќ͓1Ϫnˆ͑r͒nˆ͑r͔͒, ͑3͒ relationships in anisotropic complex fluids, as well as funda- ␧ ϭ␧ mental information about the optical properties and optical when r falls within a tube and ˜ (r) s1 otherwise. Intro- Ϫ anisotropy of the nanotubes themselves. ducing the spatial average ͗˜␧͘ϭV 1͐dr˜␧(r), where V is the In this paper, we present a semimacroscopic calculation sample volume, we obtain the intrinsic part of the dielectric of the complex dielectric tensor of carbon nanotube suspen- tensor ␧ ϭ ␧ ϭ␾ ␣ ϩ␣ Ϫ ͒ ϩ Ϫ␾͒␧ ˜ 0 ͗˜͘ ͓ ʈ͗nˆ lnˆ l͘ Ќ͑1 ͗nˆ lnˆ l͘ ͔ ͑1 s1, a͒Electronic mail: [email protected] ͑4͒

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Multiplying by eϪik"r to remove the harmonic spatial dependence and taking the spatial average, this term averages to zero,8 and the first non-vanishing correction to D is thus second order in ␦˜␧(r), being ␦˜␧͑rЈ͒ ͵ Ј ͑ Ј͒ ЈÃ Јϫ ͵ Љ ͑ Ј Љ͒ ЉÃ Љ dr G r,r ␧ dr G r ,r “ “ s • “ “ FIG. 1. A cartoon showing an aligned nanotube suspension. The suspending ␧ medium is of uniform dielectric constant s . The incident field, E0(r,t), is ␦˜␧͑rЉ͒ linearly polarized along the ‘‘slow’’ axis of the suspension. After transmis- ϫ Љ ͑ ͒ ␧ D0 , 8 sion, the polarization is unchanged but the amplitude is attenuated by ab- s • sorption. An incident field linearly polarized along the ‘‘fast’’ axis, or at 90° to the shown scenario, is altered in a similar manner. Light linearly polarized which becomes at an angle between these directions will also emerge attenuated, but with a ␦˜␧͑rЈ͒ ␦˜␧͑rЉ͒ polarization that is elliptical. We seek to calculate the effective dielectric ͵ Ј ͵ Љ ik"rЉ ␧ ͓ Љ Љ dr ␧ dr e ␧ ˜ 0 E0 tensor of the suspension. s • s • • • “ “ ͒ ͑ Ϫٌ͑2͒Љ ͔ ͑ Ј Љ͒ ͓ Ј ЈϪٌ͑2͒Ј ͔ ͑ Ј͒ 1 G r ,r • “ “ 1 G r,r . 9 8 Ϫ Resolving the singularities in the Green’s functions, intro- where ␾ is the volume fraction and ͗nˆ nˆ ͘ϭN 1͚ nˆ nˆ .As l l l l l ducing their Fourier representation, multiplying by eϪik"r in a previous derivation of the light-scattering amplitude,5 we and taking the spatial average, the correction to ˜␧ is neglect contributions of linear and higher order in the differ- 0 ence between the end-to-end vector of the lth nanotube and ͗␦˜␧2͘ 1 ˜␧ ϭ ˜␧ ϩ ͵ dq˜T͑kϪq͒ ˜␧ its body director, nˆ l , which is equivalent to assuming that the 1 ␧2 • 0 ͑2␲͒3 • 0 tubes are nominally straight. The results will be generalized s to include deformation. Ϫqqϩk21 ͫ ͬ, ͑10͒ • q2Ϫ͑kϩi␵͒2 where the tensor III. SPATIAL FLUCTUATIONS ˜ ͑ ͒ϭ ͵ Ϫiq"r͗␦␧͑ ͒ ␦␧͑ ͒͘ ␧2 ͑ ͒ The earlier derivation gives just the leading-order part of T q dre ˜ r • ˜ 0 / s 11 the complex dielectric tensor. In general, there will be is related to the structure factor. Equation ͑10͒ is analogous higher-order contributions arising from the spatial variations to the expression derived by Onuki and Doi6 and is quite ␦˜␧(r)ϭ˜␧(r)Ϫ͗˜␧͘. Following Jackson’s treatment of Ray- general. As derived here, it contains two terms of leigh scattering,7 we obtain these terms perturbatively from O(␦˜␧2/␧2); a higher-order intrinsic correction arising purely the vector wave equation for D, which can be derived from s from mean-square fluctuations and a leading-order form cor- Maxwell’s equations in a dielectric medium. In the harmonic rection involving density correlations identical to that de- approximation in which all fields have a time dependence of ␻ rived in Ref. 6. This result is somewhat intuitive, as one the form ei t, this becomes an inhomogeneous vector Helm- could envision particles lacking any spatial correlation that holtz equation could still alter the polarization of incident light, with the 2ϩ 2͒ ϭϪ Ã Ã͑ Ϫ␧ ͒ ͑ ͒ physical scenario being optically anisotropic particles tooٌ͑ k D “ “ D sE . 5 small to visibly scatter. More quantitatively, for particles Following Jackson,7 the term in parenthesis on the right-hand ͗␦␧ ␦␧ ͘ lacking any segmental correlation ˜ (r)• ˜ (0) ͑ ͒ ␦␧ ␧ ϭ␧ 2 Ϫr2/2␰2 2 3 side of Eq. 5 is written as ˜ (r) D0 / s , where D0 ˜ 0 Ϸlim␰→ ͗␦˜␧ ͘e ϰlim␰→ ͗␦˜␧ ͘␰ ␦(r), and the second ik"r • 0 0 •E0e . As in scattering theory, a formal solution is then term in Eq. ͑10͒ vanishes, implying that for such ‘‘formless’’ obtained via successive iterations in terms of the Helmholtz ␧ nanotubes the leading-order intrinsic correction to ˜ 0 is sim- Green’s function ply proportional to the mean-square deviation.9 The full ex- ͉ Ϫ Ј͉ Ϫ Ј pression for the dielectric tensor is thus 1 eik r r 1 eiq•(r r ) ͑ Ј͒ϭ ϭ ͵ 2 G r,r 3 dq 2 2 , ͗␦˜␧ ͘ 1 4␲ ͉rϪrЈ͉ ͑2␲͒ q Ϫ͑kϩi␵͒ ␧Ϸ ϩ ϩ ͵ ˜ ͑ Ϫ ͒ ͑ ͒ ˜ ͭ 1 ␧2 ͑ ␲͒3 dqT k q 6 s 2 where the positive infinitesimal ␵ is needed to ensure causal- Ϫqqϩk21 6 ͫ ͬͮ ˜␧ ϩ..., ͑12͒ ity in the displacement field, D(r,t). The first-order correc- • q2Ϫ͑kϩi␵͒2 • 0 tion is ␧ where we have exploited that fact that ˜ 0 is symmetric. Ј ͑ Јٌ͒ЈϫٌЈϫ␦␧͑ Ј͒ Ј ␧ ͵ dr G r,r ˜ r •D0/ s IV. A USEFUL EXPANSION ϭ ͵ Ј␦␧͑ Ј͒ ͑ Ј ␧ ͒ ͓ Ј ЈϪٌ͑2͒Ј ͔ ͑ Ј͒ dr ˜ r • D0/ s • “ “ 1 G r,r . Before deriving specific results applicable to limiting ͑7͒ cases and scenarios of practical importance, it is useful to

Ϫ " In this approximation, the imaginary part of Eq. ͑24͒—which ␦␧ ϰ ␣ Ϫ␣ ͒ ϩ ␣ Ϫ␧ ͒ iq rj ͑ ͒ ˜ q ͚ ͓͑ ʈ Ќ nˆ jnˆ j ͑ Ќ s 1͔e , 18 j arises from the pole structure of the integrand and, in nonabsorbing materials, gives rise to turbidity and what is the index j running over all of the scattering elements of the known as form dichroism6,11 in anisotropic systems— system. The connection to depolarized light scattering is now vanishes. Physically, this is equivalent to assuming that ab- 10 5 apparent, with each of the four structure factors being the sorption dominates scattering, where the latter scales as k4.7 square modulus of the corresponding Cartesian projection of Depending on the size of the tubes, this approximation will ␦␧ the tensor ˜ q . Following the analysis in Ref. 5, we decom- be reasonable for dilute to semidilute carbon nanotube sus- pose this tensor into inter and intratube contributions to ob- pensions. As evidence for this, we measured the absolute tain turbidity (␭ϭ632.8 nm) due to scattering in nonabsorbing N spherical colloidal suspensions prepared at a ﬁxed volume Ϫ " Ϫ3 ␦␧ ϭ ␣ Ϫ␣ ͒ ϩ ␣ Ϫ␧ ͒ iq rl ␾ Ϸ ˜ q ͚ ͓͑ ʈ Ќ nˆ lnˆ l ͑ Ќ s 1͔e fraction of 0 10 . For aqueous latex suspensions of di- lϭ1 ameter dϭ11 ␮m, 230, and 40 nm, the measured turbidity is 0.22, 0.59, and 0.12 mmϪ1, respectively, the turbidity of the Ϫ " Ј Ϫ ϫ ͵ drЈe iq r , ͑19͒ empty quartz cell being 0.07 mm 1. For all three of these vl ⌬ Ϸ Ϫ Ϸ suspensions, n/ns (1.6 1.33)/1.33 0.2, and the 2-mm- where nˆ l , rl , and vl are the director, center of mass, and thick samples were visually transparent. In contrast, a sus- volume of the lth nanotube, respectively, and rЈ locates the pension of multiwalled carbon nanotubes ͑MWNTs͒ dis- ͑ volume element drЈ with respect to the center of mass, rl . persed in toluene at the same volume fraction with a mean For reasonably monodisperse nanotubes with random cen- tube length of Lϭ10 ␮m and a mean tube diameter of 2a ϭ Ϫ1 troids, rl , we obtain 50 nm) exhibits an attenuation of 0.95 mm . In the 2-mm-thick quartz cell, the MWNT-toluene suspension had a ˜ ͑ ͒Ϸ␾␤˜ ͉͗ ͑ ͉͒2͘ ␧2 ͑ ͒ T q 1 f l q /v s , 20 black hue but was also visually transparent. The index of refraction of toluene is of order 1.5 and previous reﬂectivity where v is the average volume and measurements5 suggest that the approximate modulus of the complex MWNT refractive index is of order 1.6. Comparing ͑ ͒ϭ ͵ Ϫiq"r ͑ ͒ f l q dre 21 the measured turbidity of the MWNT suspension with that of vl the 11 ␮m and 40 nm spherical latex suspensions—which is a form amplitude that depends on the shape and orientation span the same physical dimensions and have roughly 3ϫ the of the lth tube. The earlier expression for ˜T(q) will be pre- optical contrast—we see that absorption accounts for at least cise in the dilute and semidilute limits, where the distribution 85% of attenuation in the MWNT suspension, consistent of centroids is essentially random, and when there is limited with the earlier approximation. At this volume fraction, we

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␲ ␾ 2Ϸ ␲ ␾ Ϸ note that (4/ ) 0(L/d) 50 and (4/ ) 0(L/d) 0.25, with the MWNT suspension being well into the semidilute regime.

V. LIMITING CASES By dispersing the nanotubes in a viscoelastic polymer melt or solution, flow can be used to orient the tubes, where the direction and extent of orientation depend on the type of flow, the tube aspect ratio,12 the fluid elasticity,13 and, for semidilute suspensions, hydrodynamic interactions.14,15 For dilute and semidilute suspensions in purely viscous fluids under simple steady shear flow, the distribution of fiber ori- entations is broadly peaked along a direction very close to the flow axis.12–15 In viscoelastic fluids under shear, the observed orientation is somewhat more complicated. For dilute suspensions of fibers in weakly elastic fluids, or at a small Deborah number ͑defined as the product of the shear rate and a linear-viscoelastic relaxation time͒, the steady-state orientation is along the direction of vorticity, perpendicular to the 16 FIG. 2. A cartoon showing three different scenarios of tube orientation. The plane defined by the flow and velocity-gradient directions. x axis indicates the direction of flow, and in each case the director distribu- At higher concentrations in such fluids, however, both flow tion is assumed to be symmetric around the y and z axes. An isotropic and vorticity alignment have been reported.13,16 In contrast, suspension ͑a͒ exhibits no anisotropy, while a perfectly aligned suspension for dilute and semidilute suspensions in highly elastic fluids, ͑b͒ exhibits anisotropy reflecting that of the individual nanotubes in the limit of infinite dilution. The scenario in ͑c͒ models a broad number of physically or at a high Deborah number, strong flow alignment is relevant situations. For suspensions under shear flow, tubes will be in peri- 13,17 observed, while for dilute suspensions in fluids of inter- odic Jeffery orbits around the z axis, and the director distribution function mediate elasticity, the fibers orient at well-defined angles in can be computed as either a temporal average over a moving ensemble the flow-vorticity plane.17 Although the situation is complex, containing the same tubes or as a temporal average over a fixed spatial region, with each ‘‘snap-shot’’ of the ensemble containing a new set of the most striking and relevant feature is a deviation from the tubes. stress-optical rule in its usual form, which states that the direction of greatest optical anisotropy is along the direction of principle strain, which for shear flow is at 45° in the flow-gradient plane.1,6,18 In contrast, at least for nanotubes and that can be modeled as long stiff fibers, we would expect that 2 (1) 2 (2) 2 qЌ→͓qЌ eˆЌ ͔ ϩ͓qЌ eˆЌ cos ␪ Ϫq sin ␪ ͔ , ͑28͒ in many cases the dielectric tensor will be diagonal in the • • l x l (1)ϭ ϫ ͉ ϫ ͉ (2)ϭ ϫ (1) orthogonal coordinate system defined by the flow field. The where eˆЌ nˆ l xˆ/ nˆ l xˆ and eˆЌ xˆ eˆЌ . To proceed fur- dynamic nature of the tube orbits and, at low Peclet number, ther, we note that for ease of computation ͑and to a reason- thermal fluctuations will both influence the steady-state dis- ably good approximation, particularly in the presence of tribution. polydispersity͒ we can model the square of the functions We can thus cover a broad range of physically relevant appearing in Eq. ͑26͒ with Gaussians as situations by considering the three scenarios in Fig. 2. An sin2͑x͒ J2͑y͒ 1 x2 y 2 isotropic distribution ͓Fig. 2͑a͔͒ is trivial, since there is no 1 Ϸ ͩ Ϫ Ϫ ͪ ϩ ͑ ͒ 2 2 exp ..., 29 anisotropy in Eq. ͑23͒, and the idealized scenario in Fig. 2͑b͒ x y 4 3 4 ϭ ϭ can be viewed as a limiting case of the general situation with x Llqx/2 and y alqЌ , from which we obtain depicted in Fig. 2͑c͒ in which the width of the angular dis- 2 2␺ qx L tribution tends to zero, so we focus on the distribution in Fig. ͉͗f ͑q͉͒2͘Ϸ 2ͭ 1Ϫ ͫ ϩa2͑1Ϫ␺͒ͬ ͑ ͒ l v 2 c . The form amplitude of a tube of length Ll and radius 4 3 al , centered at the origin with its director nˆ l pointing along 2 2 qЌ L the x axis, is Ϫ ͫa2͑1ϩ␺͒ϩ ͑1Ϫ␺͒ͬϩ...ͮ 8 3 ͒ ͒ sin͑Llqx/2 J1͑alqЌ f ͑q͒ϭ2v , ͑26͒ Ϸ 2 Ϫ ␰ 2Ϫ ␰ 2 ͑ ͒ l l ͒ v exp͓ ͑ ʈq ͒ ͑ ЌqЌ͒ ͔, 30 ͑Llqx/2 alqЌ x where we have introduced ␺ϭ͗cos2 ␪͘—equal to 1 for per- where J1(y) is the Bessel function of the first kind of order ϭ ϩ ϭ␲ 2 ͓ fect alignment and 1/3 for a random distribution—and we 1, qЌ qxxˆ qyyˆ, and vl al Ll . This expression Eq. ͑26͔͒ can be transformed to the general situation in which the have approximated the truncated series with a Gaussian. The ␪ correlation lengths appearing in Eq. ͑30͒ are tube axis makes an angle l with the x axis by a rotation, which leads to the substitutions 2 2 2 ␰ʈ ϭ͓L ␺/3ϩa ͑1Ϫ␺͔͒/4 ͑31͒ → ␪ ϩ (2) ␪ ͑ ͒ qx qx cos l qЌ•eˆЌ sin l 27 and

2 2 2 ␰Ќϭ͓a ͑1ϩ␺͒ϩL ͑1Ϫ␺͒/3͔/8. ͑32͒ The integrals involved in evaluating Eq. ͑25͒ are nontrivial, ˜ although by symmetry I0 is diagonal. We make one more simplifying approximation by exploiting the fact that for reasonably well-aligned tubes of sufficiently high aspect ratio, ␰ ӷ␰ ˜ Ϸ we have ʈ Ќ , in which case (I0)xx 0 and ˜ ϷϪ ␲1/2͒Ϫ1 ␰ ͒ ␰ ͒2 ϩ ͒ϩ ͑ ͒ I0 ͑16 ͑L/ ʈ ͑a/ Ќ ͑yˆyˆ zˆzˆ ... . 33 Finally, we note that for the ellipsoidal model we are considering FIG. 3. The geometry of depolarized flow light scattering, where s is the ϭ␺ ϩ 1 ͑ Ϫ␺͒͑ ϩ ͒ ͑ ͒ polarization of the incident beam and p is the polarization of the analyzer. ͗nˆ lnˆ l͘ xˆxˆ 2 1 yˆyˆ zˆzˆ . 34 For simple shear, the flow direction is along the x axis, the gradient direction All of the elements of the dielectric tensor are now cast ex- is along the y axis, and the vorticity direction is along the z axis. In previous publications ͑see Refs. 5 and 13͒, the x direction is referred to as h and the plicitly in terms of ␾, ␺, La, ␣Ќ , ␣ʈ , and ␧ . s z direction is referred to as v. We consider scattering in the x – z plane, with the four structure factors given by the four permutations of s and p. The ץ ץϭ␥ shear rate is ˙ vx / y. VI. BIREFRINGENCE AND DICHROISM

Because the parameters ␣Ќ and ␣ʈ are complex, the dielectric tensor will contain real and imaginary parts. In the 3 1 ⌬nЈϷ ͑␣Јʈ Ϫ␣ЌЈ ͒␾͑␺Ϫ ͒ϩ... ͑39͒ orthogonal coordinates defined by the principal axes of ˜␧, 4 3 ␧ ϭ ␦ one then has ˜ ␮␯ e␯ ␮␯ , where e1 , e2 , and e3 are the three and complex eigenvalues of ˜␧. The three complex refractive in- ⌬ ЉϷ 3 ͑␣ЉϪ␣Љ ͒␾͑␺Ϫ 1 ͒ϩ ͑ ͒ dices are n␯ϭn␯Јϩin␯Љϭͱe␯, with the birefringence and di- n 4 ʈ Ќ 3 ... . 40 chroism defined as the difference between the largest and The quantity (3/2) (␺Ϫ 1/3) is a nematic order parameter, smallest values of the real and imaginary parts, respectively. being zero for a random distribution and 1 for perfect align- In contrast to polymer solutions, the intrinsic dichroism of a ment. Equations ͑39͒ and ͑40͒ will only be rigorously correct carbon nanotube suspension will be significant, reflecting the in the limit of small volume fraction, and for larger ␾ one imaginary component of the polarizability, which gives rise might need higher-order corrections, evaluating nʈ and nЌ to absorption. The intensity of light polarized along the di- ␯ Ϫ ␲ Љ ␭ ␭ explicitly in terms of the real and imaginary parts of the rection will be attenuated via exp( 4 n␯d/ ), where is diagonal elements of the dielectric tensor. In principle, how- the wavelength in air and d the propagation distance. Split- ␯ ␧ ever, a direct measure of the optical anisotropy of nanotubes ting the th eigenvalue of ˜ into real and imaginary parts as can be determined from rheo-optical data if the order param- ϭ Јϩ Љ e␯ e␯ ie␯ , one obtains the exact relations eter can be measured independently. Conversely, if the opti- 2 2 1/2 1/2 e␯Јϩ͓͑e␯Ј͒ ϩ͑e␯Љ͒ ͔ cal anisotropy of the tubes is known, the order parameter can nЈϭͭ ͮ ͑35͒ ␯ 2 be determined from a measurement of either the birefringence or the dichrosim. Analogous versions of Eqs. ͑39͒ and and ͑40͒ are derived via alternate approaches by Fuller1 and Eq. 2 2 1/2 1/2 ͑39͒ is recently derived in the context of rod-like particle Ϫe␯Јϩ͓͑e␯Ј͒ ϩ͑e␯Љ͒ ͔ nЉϭͭ ͮ , ͑36͒ suspensions by Lenstra, Dogic, and Dhont.19 ␯ 2 ␯ϭ ϭ ␧ where 1, 2, 3. For the model derived here, eʈ (˜ )xx and eЌϭ(˜␧) ϭ(˜␧) , with the birefringence and dichroism yy zz VII. CONNECTION TO LIGHT SCATTERING ⌬nЈϭnЈʈ ϪnЌЈ and ⌬nЉϭnЉʈ ϪnЌЉ , respectively. Comparison with experiment might require a numerical If the tubes have no optical anisotropy, then the light- ͉ ͉2 computation of the earlier quantities, but for the sake of il- scattering intensity is simply proportional to ͗ f l(q) ͘. Oth- lustration we explicitly consider only the leading-order con- erwise, for a given scattering geometry ͑Fig. 3͒ there are four tribution here, noting that without some type of small- structure factors contrast approximation, the general expressions become ͑ ͒ϰ͉ ␦␧ ͉2 quite involved. Then S␮␯ q xˆ␮• ˜ q•xˆ␯ ͱ 1 nʈϭ eʈϷ␧ ϩ ␾␺͓͑␣Јʈ Ϫ␣Ј ͒ϩi͑␣Љʈ Ϫ␣Љ ͔͒ϩ... ͑37͒ s 2 Ќ Ќ ϭ͚ͯ ͓͑␣ʈϪ␣ ͒ Ќ xˆ␮ nˆ lnˆ l xˆ␯ l • • and 2 Ϫ " ϭͱ Ϸ␧ ϩ 1 ␾͑ Ϫ␺͓͒͑␣ЈϪ␣Ј ͒ϩ ͑␣ЉϪ␣Љ ͔͒ ϩ͑␣ Ϫ␧ ͒␦ ͔ ͑ ͒ iq rlͯ ͑ ͒ nЌ eЌ s 4 1 ʈ Ќ i ʈ Ќ Ќ s ␮␯ f l q e . 41 ϩ..., ͑38͒ As in Sec. IV, for a random distribution of nanotube cen- from which troids this reduces to

Downloaded 21 Jul 2004 to 129.6.154.32. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp 1034 J. Chem. Phys., Vol. 121, No. 2, 8 July 2004 Erik K. Hobbie

͒ϰ ͉ ␣ Ϫ␣ ͒ S␮␯͑q ͚ ͑ ʈ Ќ xˆ␮ nˆ lnˆ l xˆ␯ l • • ϩ ␣ Ϫ␧ ͒␦ ͉2͉ ͉͒2 ͑ ͒ ͑ Ќ s ␮␯ f l͑q . 42 In the decoupling approximation Eq. ͑22͒, the four structure factors have the same anisotropy. Although this approximation might be appropriate at the level of computing the form birefringence and dichroism—particularly in light of the strong intrinsic terms—simple consideration of S␮␯(q) shows that it is not appropriate for scattering; ˜␧ will always be isotropic for random distributions but S␮␯(q) is by deﬁ- nition anisotropic. We thus require a more exact coupling scheme, and the following approach can be used to reﬁne Eq. ͑20͒. Written out explicitly, the three independent structure factors in the x – z plane are

͑ ͒ϭ Ϫ1␧Ϫ2 ͓͉␦ ͉2ϩ ͑␦ ␦ ͒ 2 ␪ Sxx q V s ͚ 1 2Re 1* 2 cos l l ϩ͉␦ ͉2 4 ␪ ͉ ͉͒2 ͑ ͒ 2 cos l͔ f l͑q , 43

͑ ͒ϭ Ϫ1␧Ϫ2 ͓͉␦ ͉2ϩ ͑␦ ␦ ͒ 2 ␪ 2 ␸ Szz q V s ͚ 1 2Re 1* 2 sin l cos l l ϩ͉␦ ͉2 4 ␪ 4 ␸ ͉ ͉͒2 ͑ ͒ 2 sin l cos l͔ f l͑q , 44 and ͒ϭ ͒ Sxz͑q Szx͑q

ϭ Ϫ1␧Ϫ2͉␦ ͉2 2 ␪ 2 ␪ 2 ␸ ͉ ͑ ͉͒2 V s 2 ͚ cos l sin l cos l f l q , l ͑45͒ ␦ ϭ␣ Ϫ␧ ␦ ϭ␣ Ϫ␣ where 1 Ќ s , 2 ʈ Ќ , and f l(q) is given by Eq. ͑26͒ with the substitutions ͑27͒ and ͑28͒. Evaluation of Eqs. ͑43͒–͑45͒ is done numerically for a given orientational distribution, p(␪). Figure 4 shows the computed depolarized structure factors for an isotropic distribution of nanotubes in the Gaussian approximation ͓Eq. ͑29͔͒ with Lϭ10 ␮m, aϭ25 nm, ͉␦ ͉2 ␧2ϭ ͉␦ ͉2 ␧2ϭ ␦ ␦ ␧2ϭ 1 / s 0.3, 2 / s 0.28, and 2 Re( 1* 2)/ s 0.58. As an approximation, ␣Ќ and ␣ʈ are assumed real. Measure- FIG. 4. The numerically computed structure factors for an isotropic distri- ments on aligned metallic single-walled carbon nanotubes bution of nanotubes in the Gaussian approximation ͓Eq. ͑29͔͒, with L ϭ ␮ ϭ ͉␦ ͉2 ␧2ϭ ͉␦ ͉2 ␧2ϭ ͓␦ ␦ ͔ ␧2 ͑SWNTs͒ suggest that there is no absorption normal to the 10 m, a 25 nm, 1 / s 0.3, 2 / s 0.28, and 2 Re 1* 2 / s ϭ ␮ Ϫ1 tube axis, with strong absorption parallel to the tube axis.20 0.58. The width of each scattering pattern is 3.5 m . These parameters are chosen to reflect the physical dimensions of the MWNTs used for scat- For MWNTs, which typically have a larger number of de- tering and microscopy measurements ͑Fig. 5͒, with the optical anisotropy fects in their graphitic structure, this absorption anisotropy adjusted to obtain agreement with the data in Fig. 5. will be present but less pronounced, and we neglect it here as a first approximation. A comparison between experiment and ␣ʈ/␣ЌϷ1.4, somewhat lower than the lower limit of scat- theory that accounts for absorption and incorporates a variety tering anisotropy measured for aligned SWNT bundles.21 of rheo-optical measurements will be reported elsewhere. The approximate qϪ1 scattering envelope in Fig. 6 arises The calculations reproduce the distinctive features of mea- from the tube-like structure of the MWNTs. For a semiflex- sured depolarized light-scattering patterns under comparable ible chain of arbitrary orientation, the radial density of scat- conditions ͑Fig. 5͒, and a quantitative study of the effect of tering elements is n(r)ϰrD, where Dу1 is the fractal di- ␧ Ϸ varying anisotropy is shown in Fig. 6. Assuming s 1.5, mension of the chain. The radial distribution function is thus ␦ ␦ Ϸ ␣ ␣ Ϸ 1Ϫd DϪd ϪD . rϰr , from which S(q)ϰqץ/nץ implies ʈ / Ќ 3, somewhat higher than the g(r)ϰr 1.83 1 / 2 upper limit of scattering anisotropy measured for aligned 21 ␦ ␦ ͑ VIII. TUBE DEFORMATION SWNT bundles. Increasing 1 with respect to 2 or, equivalently, decreasing ␣ʈ/␣Ќ) gives good agreement be- The micrographs in Fig. 5 suggest that the nanotubes are tween the MWNT data and theory, with the data suggesting not ideally straight. Tube deformation can arise from a vari-

FIG. 6. The depolarized structure factor Szz(q) computed for varying anisotropy, where the dashed line shows qϪ1 power-law behavior. Scattering proﬁles have been generated for an isotropic distribution of Lϭ1 ␮m, a ϭ ͑ ͒ ␦ ␦ Ϸ 1 nm SWNTs upper curve assuming 2 / 1 1.83, and for isotropic dis- ϭ ␮ ϭ ␦ ␦ Ϸ tributions of L 10 m, a 25 nm MWNTs assuming 2 / 1 1.83, 1.3, and 1.03, where the latter gives the best agreement with scattering data in Fig. 5.

Using the decomposition ϭ ␨ ͒Ϫ␨ ͒ ϩ␨ ͒ ͑ ͒ ͗nˆ jnˆ j͘l ͓ ʈ͑l Ќ͑l ͔nˆ lnˆ l Ќ͑l 1, 48

where the body director nˆ l is the dominant eigenvector of ␨ ␨ ͗nˆ jnˆ j͘l with eigenvalue ʈ(l) and Ќ(l) is the remaining

FIG. 5. Depolarized light-scattering patterns measured for an isotropic semidilute (␾Ϸ3ϫ10Ϫ3) MWNT-polymer suspension with a mean tube length and diameter comparable to those used in Fig. 4 (LϷ10 ␮m, a Ϸ25 nm). The width of the upper right scale bar is 3.5 ␮mϪ1. The lower image is a 200ϫ optical micrograph of the same suspension, with a scanning electron microscopy image before dispersion as inset (width ϭ1.3 ␮m). The scale bar is 10 ␮m. Due to absorption, individual tubes appear as shadows on a bright incoherent background, in much the same way that incoherent light emitted on a dark background creates an image of a ﬂuorescently labeled deoxyribose nucleic acid ͑DNA͒ molecule. As in the case of ﬂuorescently labeled DNA, the MWNTs appear to be of larger diameter than they actually are.

ety of sources, including structural defects ͑in which case the deformation is ‘‘quenched’’ onto the tube͒, the flexure of extremely long suspended tubes due to thermal fluctuations in solution, and applied and residual hydrodynamic stresses acting on tubes in flowing suspensions and melts. The results derived in the previous sections can be generalized to include tube deformation via a renormalization of the dielectric an-

isotropy, ␣ʈ and ␣Ќ , and a generalization of the form amplitude, f l(q). As shown in Fig. 7, we can model the lth nano- ϭ tube as Nl consecutive steps rj͕l͖ bnˆ j͕l͖, where b is the ϭ segment length and Ll bNl is the contour length. The per- ␰ sistence length of the tube can then be deﬁned as p(l) Ϸ ͓ Ϫ͗ ͘ ͔Ϫ1 ͗ ͘ b 1 nˆ j•nˆ jϩ1 l , where the brackets ... l denote an internal conformational average over the lth nanotube. The local dielectric tensor of the jth element of this tube is given FIG. 7. To include deformation, we model a tube as the series of consecu- ϭ ϭ by Eq. ͑3͒: tive steps, rj bnˆ j , where b is the segment length and L bN is the contour length. The body director, nˆ, is the dominant eigenvector of the orientation ͑␣ Ϫ␣ ͒ ϩ␣ ͑ ͒ ͗ ͘ ͗ ͘ ʈ Ќ nˆ jnˆ j Ќ1, 46 tensor, nˆ jnˆ j , and the moment tensor, rr , where the vector r locates an element of the tube with respect to its centroid. The eigenvalues of the with the mean dielectric tensor of the tube given by former renormalize the optical anisotropy, while the eigenvalues of the latter deﬁne the ellipsoidal Gaussian envelope that gives a measure of the effec- ␣ Ϫ␣ ͒ ϩ␣ ͑ ͒ ͑ ʈ Ќ ͗nˆ jnˆ j͘l Ќ1. 47 tive shape.

Downloaded 21 Jul 2004 to 129.6.154.32. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp 1036 J. Chem. Phys., Vol. 121, No. 2, 8 July 2004 Erik K. Hobbie smaller ͑azimuthally-symmetric͒ eigenvalue, Eq. ͑47͒ be- N 2 comes 2ϭ͗ 2͘ϭͳ ͩ ͚ ͪ ʹ ͑ ͒ Rg R b nˆ j 57 jϭ1 ␣ Ϫ␣ ͒ ␨ ͒Ϫ␨ ͒ ϩ ␣ ϩ␨ ͒ ␣ Ϫ␣ ͒ ͑ ʈ Ќ ͓ ʈ͑l Ќ͑l ͔nˆ lnˆ l ͓ Ќ Ќ͑l ͑ ʈ Ќ ͔1, ͑49͒ is the mean-square end-to-end vector of the chain, which is simply related to the dominant eigenvalue of ͗rr͘ defined ͗ ͘ϭ ␨ ϩ␨ ϭ ͑ ͒ where Tr nˆ jnˆ j 2 Ќ ʈ 1. Comparing Eq. 49 with the earlier, and Eq. ͑57͒ can be evaluated in the limit of large N ͓␣eff coarse-grained expression for the same quantity, ʈ (l) to obtain Ϫ␣eff ϩ␣eff Ќ (l)͔nˆ lnˆ l Ќ (l)1, we obtain expressions for the renor- Nb2 sinh͑b/␰ ͒ malized anisotropy of the lth nanotube 2Ϸ p Ϸ ␰ ϭ ␰ ͑ ͒ Rg ͑ ␰ ͒Ϫ 2Nb p 2L p . 58 eff cosh b/ p 1 ␣Ќ ͑l͒ϭ␣Ќϩ␨Ќ͑l͒͑␣ʈϪ␣Ќ͒ ͑50͒ The remaining eigenvalue follows from the mean-square and transverse displacement eff eff N ␣ʈ ͑l͒Ϫ␣Ќ ͑l͒ϭ͑␣ʈϪ␣Ќ͓͒␨ʈ͑l͒Ϫ␨Ќ͑l͔͒. ͑51͒ 2 ϭ Ϫ1 ͳ ͚ ͓ Ϫ͑ ͒ ͔2 ʹ ͑ ͒ RЌ N roj roj nˆ nˆ , 59 The results of the preceding sections can be generalized via jϭ1 • ␣ → ␣eff ␣ → ␣eff the substitutions ʈ ͗ ʈ (l)͘ and Ќ ͗ Ќ (l)͘, where the ϭ ͚ j where roj b iϭ1nˆ i locates the jth segment of the tube with brackets denote an ensemble average over the suspension. respect to one of its ends. For stiff chains in which the ma- The second generalization involves the form amplitude, jority of segments lie on R, this can be approximated by ͑ ͒ f l(q), as given by Eq. 21 . Written out explicitly, this is N 2 Ϸ 2 Ϫ1 ͗␪2͘ϭ 3 ␰ ͑ ͒ Ϫ " 1 RЌ b N ͚ j b / p , 60 ͑ ͒ϭ ͵ iq rϷ ͵ ͫ Ϫ " Ϫ ͑ " ͒2ϩ ͬ ϭ f l q dre dr 1 iq r q r ... , j 2 vl vl 2 ͑52͒ where the second step follows from equipartition. Note that RЌ vanishes in the limit of large ␰ , as expected. In a similar in the limit of small q. Integrating Eq. ͑52͒ term by term, the p manner, it is straightforward to show that for ͗nˆ nˆ ͘, ␨ʈ linear contribution vanishes, giving the Gaussian approxima- j j Ϸ(2/N)(R /b)Ϫ1 and ␨ЌϷ1Ϫ(1/N)(R /b). tion g g 1 1 IX. CONCLUSIONS f ͑q͒ϭv ͩ 1Ϫ q ͗rr͘ qϩ...ͪ Ϸv expͩ Ϫ q ͗rr͘ qͪ , l l 2 • l• l 2 • l• We have derived an expression for the dielectric tensor ͑53͒ of a dilute to semidilute suspension of optically anisotropic nanotubes. The result is expressed as a power series in the which depends on the moment tensor, ͗rr͘l Ϫ volume fraction, optical anisotropy, and optical contrast of ϭv 1͐ dr(rr). By symmetry, this will have the same prin- l vl ͗ ͘ the suspension. Within this framework, the connection to de- cipal axes as nˆ jnˆ j l , the dominant eigenvector being the polarized light scattering has been established and the results body director, nˆ l . Denoting the large eigenvalue 2 2 have been generalized to include tube deformation of arbi- (1/12) Leff(l) and the small eigenvalue (1/4) aeff(l), a tube trary origin. A variety of rheo-optical data ͑light-scattering, with its body director aligned with the x axis has optical microscopy, birefringence, and dichrosim͒ for flow- 1 1 ing polymer-dispersed carbon nanotube suspensions will be ͉f ͑q͉͒2Ϸv2 exp͓Ϫ L2 ͑l͒q2Ϫ a2 ͑l͒q2 ͔, ͑54͒ l l 12 eff x 4 eff Ќ analyzed elsewhere. By incorporating an independent mea- which when combined with Eqs. ͑27͒ and ͑28͒ is sufficient to sure of the shape of the tubes and the orientational distribu- include tube deformation in the Gaussian approximation, all tion function, p(␪) ͑from stroboscopic video microscopy for of the previous results being generalized via the substitutions MWNTs or small-angle neutron scattering for SWNTs, for 2→͗ 2 ͘ 2→͗ 2 ͘ ͒ L Leff and a aeff . example , we expect to obtain quantitative information about As a specific example we consider the Kratky–Porod the optical anisotropy of carbon nanotubes that can then be model of a semiflexible chain with no torsional stress,22 with used to quantify their orientation in flowing viscoelastic the Hamiltonian polymer melts. Although the results are derived in the dilute and semidilute limits, the results can be applied to more con- ␬ N ␬ N 3ϭ ␲ ␾ 2Ͼ ϭϪ ϭϪ ␪ ͑ ͒ centrated suspensions, for which nL (4/ ) 0(L/d) 1 H ͚ nˆ j nˆ jϪ1 ͚ cos j , 55 2ϭ ␲ ␾ Ͼ b jϭ2 • b jϭ2 and ndL (4/ ) 0(L/d) 1, by including interparticle cor- Ϫ relations in a derivation of an effective form contribution that ␪ ϭ 1 23 where j cos (nˆ j•nˆ jϩ1) is the bending angle between suc- incorporates structure. cessive orientation vectors, ␬ is the bending modulus, and We conclude by noting the paucity of optical anisotropy ␰ ϭ␬ the persistence length is p /kBT. The orientational corre- data on carbon nanotubes. In part, this reflects the scarcity of lation function decays exponentially with distance along the well-dispersed, well-aligned nanotube materials. For the sole 22 chain via purpose of measuring this anisotropy, elongational flows will Ϫ ͉ Ϫ ͉ ␰ yield better fiber alignment than shearing flows, and one ͗nˆ nˆ ͘ϭe b i j / p. ͑56͒ i• j could envision spinning composite fibers in noncrystalline We assume that the chain is stiff, so that body director nˆ is polymer matrices that could then be optically probed at a approximately equivalent to R/Rg , where variety of wavelengths to map out the full dielectric spec-

Downloaded 21 Jul 2004 to 129.6.154.32. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp J. Chem. Phys., Vol. 121, No. 2, 8 July 2004 Optical anisotropy of nanotube suspensions 1037 trum. All of the earlier results would directly apply to the optically anisotropic particles too small to scatter light. The second term in analysis of such data. The formalism derived here might also Eq. ͑12͒ does not appear in the analysis of Onuki and Doi ͓A. Onuki and ͑ ͔͒ prove useful in optical flow studies of any type of dilute or M. Doi, J. Chem. Phys. 85, 1190 1986 because it is of higher order in products of the optical contrast, anisotropy, and volume fraction when semidilute rod-like particle or fiber suspension, such as actin there is segmental correlation, which will be quite strong for carbon nano- filaments, tobacco mosaic virus, worm-like micelles, and he- tubes. We retain it here for a theoretical description that is rigorous to matite rods,24,25 in particular, which should also exhibit in- O(␦˜␧2). 10 trinsic absorption. The transformation from the notation used in E. K. Hobbie, H. Wang, H. Kim, C. Han, E. Grulke, and J. Obrzut ͓see Ref. 5͔ to that used here is ␣ → ␣ Ϫ␧ ␧ ␣ → ␣ Ϫ␧ ␧ made via the substitutions ʈ ( ʈ s)/ s and Ќ ( Ќ s)/ s in the ACKNOWLEDGMENTS previously derived expressions for the depolarized structure factors ͓Ref. 5͔. In the present notation, the previously defined scattering parameters ␣ The author thanks D. Fry and R. K. Hobbie for useful ␦ ␣ϭ ␣ ϩ ␣ Ϫ␧ ␧ ␦ϭ ␣ Ϫ␣ ␧ and are ͓1/3 ( ʈ 2 Ќ) s͔/ s and ( ʈ Ќ)/ s . discussions, E. A. Grulke for providing the scanning electron 11 N. J. Wagner, G. G. Fuller, and W. B. Russel, J. Chem. Phys. 89, 1580 microscopy image in Fig. 5, and H. Wang and H. Kim for ͑1988͒. assistance with the optical measurements and useful discus- 12 E. J. Hinch and L. G. Leal, J. Fluid Mech. 52,683͑1972͒. sions. 13 See, for example, E. K. Hobbie, H. Wang, H. Kim, S. Lin-Gibson, and E. A. Grulke, Phys. Fluids 15, 1196 ͑2003͒, and references therein. 14 C. A. Stover, D. L. Koch, and C. Cohen, J. Fluid Mech. 238,277͑1992͒. 1 See, for example, G. G. Fuller, Optical Rheometry of Complex Fluids 15 M. Rahnama, D. L. Koch, and E. S. G. Shaqfeh, Phys. Fluids 7,487 ͑Oxford University Press, New York, 1995͒, and references therein. ͑1995͒. 2 M. Stoimenova, L. Labaki, and S. Stoylov, J. Colloid Interface Sci. 77,53 16 Y. Iso, D. L. Koch, and C. Cohen, J. Non-Newtonian Fluid Mech. 62,115 ͑ ͒ 1980 . ͑1996͒. 3 ͑ ͒ A. Peterlin and H. A. Stuart, Z. Phys. 112,1 1939 . 17 Y. Iso, C. Cohen, and D. L. Koch, J. Non-Newtonian Fluid Mech. 62,135 4 See, for example, R. H. Baughman, A. A. Zakhidov, and W. A. de Heer, ͑1996͒. ͑ ͒ Science 297,787 2002 , and reference therein. 18 Deviations from the stress-optical rule have also been reported in 5 E. K. Hobbie, H. Wang, H. Kim, C. Han, E. Grulke, and J. Obrzut, Rev. branched polymer melts ͓S. B. Kharchenko and R. M. Kannan, Macro- ͑ ͒ Sci. Instrum. 74, 1244 2003 . molecules 36, 407 ͑2003͔͒. 6 ͑ ͒ A. Onuki and M. Doi, J. Chem. Phys. 85, 1190 1986 . 19 T. A. J. Lenstra, Z. Dogic, and J. K. G. Dhont, J. Chem. Phys. 114, 10151 7 J. D. Jackson, Classical Electrodynamics ͑Wiley, New York, 1975͒. ͑ ͒ 8 2001 . As a general rule, one must make the singularities explicit when evaluat- 20 Z. M. Li, Z. K. Tang, H. J. Liu, N. Wang, C. T. Chan, R. Saito, S. Okada, ing integrals of Green’s functions. Here, this is done via the substitution G. D. Li, J. S. Chen, N. Nagasawa, and S. Tsuda, Phys. Rev. Lett. 87, ϭϪ␦ Ϫ 2 2ٌ G(r) (r) k G(r) and one must also exploit the fact that E0•k 127401 ͑2001͒. ϭ 0. 21 Z. Yu and L. Brus, J. Phys. Chem. B 105, 1123 ͑2001͒. 9 ␦ ϭ ␲ Ϫ3/2␰Ϫ3 Ϫ 2 ␰2 22 We have used the relation (r) lim␰→0(2 ) exp( r /2 ) to ob- See, for example, T. Strick, J. Allemand, V. Croquette, and D. Bensimon, tain an expression for a correlation function of vanishing spatial extent Prog. Biophys. Mol. Biol. 74,115͑2000͒, and references therein. and finite amplitude. This limit is only applicable if ␰ goes to zero while 23 J. W. Bender and N. J. Wagner, J. Colloid Interface Sci. 172,171͑1995͒. ⌳ 24 the lower cutoff wavelength, 0 , is fixed, where the latter quantity sets S. J. Johnson and G. G. Fuller, J. Colloid Interface Sci. 124,441͑1988͒. ϭ ␲ ⌳ 25 ͑ ͒ the upper cutoff wave vector qc 2 / 0 . Physically, this corresponds to S. J. Gason, D. V. Boger, and D. E. Dunstan, Langmuir 15, 7446 1999 .

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